Numerical Inversion of Travel Time Observations by Boundary Value Problem Methods

Abstract

The paper considers the inverse seismic problem of identifying geological properties of a medium from travel time observations between sources and receivers. The ray-tracing approach leads to the problem of estimating parameters in a large system of ODE boundary value problems with interior point conditions and points of discontinuity.

In recent years generalized Gauss Newton methods were developed that solve constrained nonlinear estimation problems subject to ODE systems discretized, e.g. by multiple shooting or collocation. The treatment of real-life problems from different areas shows their reliability and wide applicability.

Typical for the kind of problems discussed in this paper, occuring not only in geophysics but also in chemistry, biology, ecology or medicine, is their very large dimension. This requires special numerical methods that take advantage of the structures appearing in the dynamical model. The basic details of these methods will be presented in this paper. Using the corresponding program package PARMEX, developed by Reiners and Schloder, some examples have been treated which exhibit typical properties of real-life problems, e.g. ill-conditioning and noisy data.